We review the relations between Jordan cells in various branches of physics, ranging from quantum mechanics to massive gravity theories. Our main focus is on holographic correspondences between critically tuned gravity theories in Anti-de Sitter space and logarithmic conformal field theories in various dimensions. We summarize the developments in the past five years, include some novel generalizations and provide an outlook on possible future developments.
We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of entanglement wedge cross section. We argue that in AdS3/CFT2, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has the minimal path-integral complexity. We confirm this claim in several examples.
We develop a framework for studying the well-known universal term in the Renyi entropy for an arbitrary entangling region in four-dimensional conformal field theories that are holographically dual to gravitational theories. The shape dependence of the Renyi entropy $S_n$ is described by two coefficients: $f_b(n)$ for traceless extrinsic curvature deformations and $f_c(n)$ for Weyl tensor deformations. We provide the first calculation of the coefficient $f_b(n)$ in interacting theories by relating it to the stress tensor one-point function in a deformed hyperboloid background. The latter is the…
The near-horizon conformal symmetry of nonextremal black holes is shown to be a mandatory ingredient for the holographic scaling of the scalar-field contribution to the black hole entropy. This conformal tightness is revealed by semiclassical first-principle scaling arguments through an analysis of the multiplicative factors in the entropy due to the radial and angular degrees of freedom associated with a scalar field. Specifically, the conformal SO(2,1) invariance of the radial degree of freedom conspires with the area proportionality of the angular momentum sums to yield a robust holographic…
In this paper, we have considered a cosmological model with the non--minimal kinetic coupling terms and investigated its cosmological implications with respect to the logarithmic entropy-- corrected holographic dark energy (LECHDE). The correspondence between LECHDE in flat FRW cosmology and the phantom dark energy model with the aim to interpret the current universe acceleration is also examined.
We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.
We demonstrate that for general conformal field theories (CFTs), the entanglement for small perturbations of the vacuum is organized in a novel holographic way. For spherical entangling regions in a constant time slice, perturbations in the entanglement entropy are solutions of a Klein-Gordon equation in an auxiliary de Sitter (dS) spacetime. The role of the emergent time-like direction in dS is played by the size of the entangling sphere. For CFTs with extra conserved charges, e.g., higher spin charges, we show that each charge gives rise to a separate dynamical scalar field in dS.
In this paper, we have considered $F(R,T)$ gravity as a linear function of the curvature and torsion scalars and interacted it with logarithmic entropy corrected holographic dark energy to evaluate cosmology solutions. The model has been investigated by FRW metric, and then the energy density and the pressure of dark energy have calculated. Also we obtained equation of state (EoS) parameter of dark energy and has plotted it with respect to both variable of cosmic time and $e$-folding number. Finally, we have described the scenario in three status: early, late and future time by $e$-folding num…
We compute holographic one- and two-point functions of critical higher-curvature gravity in four dimensions. The two most important operators are the stress tensor and its logarithmic partner, sourced by ordinary massless and by logarithmic non-normalisable gravitons, respectively. In addition, the logarithmic gravitons source two ordinary operators, one with spin-one and one with spin-zero. The one-point function of the stress tensor vanishes for all Einstein solutions, but has a non-zero contribution from logarithmic gravitons. The two-point functions of all operators match the expectations …
Holographic Renormalization Group (RG) in nine dimensions is considered. The d8 holographic conformal anomaly is found. It should correspond to d8 CFT in AdS_9/CFT_8 correspondence. The comparison of holographic and QFT anomalies in d8 de Sitter space is done. It may give the indication for rigorous AdS_9/CFT_8 correspondence proposal.
In the presence of boundaries the integrated conformal anomaly is modified by the boundary terms so that the anomaly is non-vanishing in any (even or odd) dimension. The boundary terms are due to extrinsic curvature whose exact structure in $d=3$ and $d=4$ has recently been identified. In this note we present a holographic calculation of those terms in two different prescriptions for the holographic description of the boundary CFT. We stress the role of supersymmetry when discussing the holographic description of ${\cal N}=4$ SYM on a $4$-manifold with boundaries.
Using a uniformization map we determine the holographic entanglement entropy for states of a Warped Conformal Field Theory dual to a generic vacuum metric in AdS$_3$ gravity with Compère--Song--Strominger boundary conditions. We point out how that expression could lead to inequalities that can be interpreted as quantum energy conditions for Warped Conformal Field Theories.
We study CFT2 conformal blocks on a torus and their holographic realization. The classical conformal blocks arising in the regime where conformal dimensions grow linearly with the large central charge are shown to be holographically dual to the geodesic networks stretched in the thermal AdS bulk space. We discuss the n-point conformal blocks and their duals, the 2-point case is elaborated in full detail. We develop various techniques to calculate both quantum and classical conformal block functions. In particular, we show that exponentiated global torus blocks reproduce classical torus blocks …
The entanglement and Rényi entropies for spherical entangling surfaces in CFTs with gravity duals can be explicitly calculated by mapping these entropies first to the thermal entropy on hyperbolic space and then, using the AdS/CFT correspondence, to the Wald entropy of topological black holes. Here we extend this idea by taking into account corrections to the Wald entropy. Using the method based on horizon symmetries and the asymptotic Cardy formula, we calculate corrections to the Wald entropy and find that these corrections are proportional to the logarithm of horizon area. With the correcte…
We study a fourth-order derivative scalar field configuration in a fixed Lifshitz background. Using an auxiliary field we rewrite the equations of motion as two coupled second order equations. We specialize to the limit that the mass of the scalar field degenerates with that of the auxiliary field and show that logarithmic modes appear. Using non-relativistic holographic methods we calculate the two-point correlation functions of the boundary operators in this limit and find evidence for a non-relativistic logarithmic conformal field theory at the boundary.
We study defects in non-relativistic conformal field theories. As in the well-studied case of relativistic conformal defects, we find that a useful tool to organize correlation functions is the defect operator expansion (dOPE). We analyze how the dOPE is implemented in theories with a holographic dual, highlighting some interesting aspects of the operator/state mapping in non-relativistic holography.
Bosonic sector of d5 gauged supergravity with specific parametrization of full scalar coset is considered (multi-dilaton gravity). Using holographic RG in the form suggested by de Boer-Verlinde-Verlinde the holographic d4 conformal anomaly (in the sector of curvature invariants) is found when bulk potential is not constant. Its comparison with the earlier calculation done in the scheme where expansion of metric and dilaton over radial coordinate of AdS space is used demonstrates scheme dependence of holographic conformal anomaly. In AdS/CFT correspondence where dilatons play role of coupling c…
We present a simple prescription for computing conformal blocks and correlation functions holographically in AdS$_3$ in terms of Wilson lines merging at a bulk vertex. This is shown to reproduce global conformal blocks and heavy-light Virasoro blocks. In the case of higher spin theories the space of vertices is in one-to-one correspondence with the space of ${\cal W}_N$ conformal blocks, and we show how the latter are obtained by explicit computations.
Based on the matching method, we explore the effects of adding an external magnetic field on the $s$-wave holographic superconductor when the gauge field is in the form of the logarithmic nonlinear source. First, we obtain the critical temperature as well as the condensation operator in the presence of logarithmic nonlinear electrodynamics and understand that they depend on the nonlinear parameter $b$. We show that the critical temperature decreases with increasing $b$, which implies that the nonlinear gauge field makes the condensation harder. Then, we turn on the magnetic field in the bulk a…
We study a general D-dimensional Schwarzschild-type black brane solution of the Einstein-dilaton theory and derive, by using the holographic renormalization, its thermodynamics consistent with the geometric results. Using the membrane paradigm, we calculate the several hydrodynamic transport coefficients and compare them with the results obtained by the Kubo formula, which shows the self-consistency of the gauge/gravity duality in the relativistic non-conformal theory. In order to understand more about the relativistic non-conformal theory, we further investigate the binding energy, drag force…
In this note we study CFT$_2$ Virasoro conformal blocks with heavy operators in the large-$c$ limit in the context of AdS$_3$/CFT$_2$ correspondence. We compute the lengths of the holographic Steiner trees dual to the $5$-point and $6$-point conformal blocks using the superlight approximation when one or more dimensions are much less than the others. These results are generalized for $N$-point holographic Steiner trees dual to $(N+1)$-point conformal blocks with superlight weights.
Using the proposal for holographic entanglement entropy in higher derivative gravities, we compute holographic entanglement entropy for the conformal gravity in four dimensions which turns out to be finite. However, if one subtracts the contribution of the four dimensional Gauss-Bonnet term, the corresponding entanglement entropy has a divergent term and indeed restricted to an Einstein solution of the conformal gravity, the resultant entanglement entropy is exactly the same as that in the Einstein gravity. We will also make a comment on the first law of the entanglement thermodynamics for the…
We study the application of our recent holographic entanglement negativity conjecture for mixed states of adjacent subsystems in conformal field theories with a conserved charge. In this context we obtain the holographic entanglement negativity for zero and finite temperature mixed state configurations in $d$-dimensional conformal field theories dual to bulk extremal and non extremal charged $AdS_{d+1}$ black holes. Our results conform to quantum information theory expectations and constitute significant consistency checks for our conjecture.
Logarithmic operators and logarithmic conformal field theories are reviewed. Prominent examples considered here include c=-2 and c=0 logarithmic conformal field theories. c=0 logarithmic conformal field theories are especially interesting since they describe some of the critical points of a variety of longstanding problems involving a two dimensional quantum particle moving in a spatially random potential, as well as critical two dimensional self avoiding random walks and percolation. Lack of classification of logarithmic conformal field theories remains a major impediment to progress towards …
We study 2-dimensional Logarithmic Galilean Conformal Algebra (LGCA) by making use of a contraction of Topologically Massive Gravity at critical point. We observe that using a naive contraction at the critical point fails to give a well defined theory, though contracting the theory while we are approaching the critical point leads to a well behaved expressions for two point functions of the energy-momentum tensors of LGCA.
Holographic mutual and tripartite information have been studied in a non-conformal background. We have investigated how these observables behave as the energy scale and number of degrees of freedom vary. We have found out that the effect of degrees of freedom and energy scale is opposite. Moreover, it has been observed that the disentangling transition occurs at large distance between sub-systems in non-conformal field theory independent of l. The mutual information in a non-conformal background remains also monogamous.
We study hydrodynamics of four-dimensional non-conformal holographic plasma with non-equal central charges $c\ne a$ at the ultraviolet fixed point. We compute equation of state, the speed of sound waves, transport coefficients (shear and bulk viscosities), and discuss causality. We study the asymptotic character of the hydrodynamic series for the homogeneous and isotropic expansion of the plasma. We perform computations for finite $c-a\ne 0$, but to leading nonvanishing order in the conformal symmetry breaking coupling.
We review holographic renormalization for non-conformal branes using the Hamilton-Jacobi formalism. We provide the tools required for the computation of the holographic dictionary, while we present some marginally new technical results regarding holographic renormalization for $Dp$-branes, $p<5$, for Einstein-Maxwell theory coupled to a dilaton. The computations are based on a recursive algorithm by Papadimitriou, that solves the radial Hamilton-Jacobi equation in the asymptotic boundary of the bulk. This paper is a short version of the Master thesis of the author written in 2014 at the Instit…
Very special $T\bar{J}$ deformations of a conformal field theory are irrelevant deformations that break the Lorentz symmetry but preserve the twisted Lorentz symmetry. We construct a holographic description of very special $T\bar{J}$ deformations. We give a holographic recipe to study the double trace as well as single trace deformations. The former is obtained from the change of the boundary condition while the latter is obtained from the change of the supergravity background.
We study dual conformal transformations of minimal area surfaces in $AdS_5 \times S^5$ corresponding to holographic smooth Wilson loops and some other related observables. To act with dual conformal transformations we map the string solutions to the dual space by means of T-duality, then we apply a conformal transformation and finally T-dualize back to the original space. The transformation maps between string solutions with different boundary contours. The boundary contours of the minimal surfaces are not mapped back to the AdS boundary, and the regularized area of the surface changes.
Cohen and Glashow introduced the notion of very special relativity as viable space-time symmetry of elementary particle physics. As a natural generalization of their idea, we study the subgroup of the conformal group, dubbed very special conformal symmetry, which is an extension of the very special relativity. We classify all of them and construct field theory examples as well as holographic realization of the very special conformal field theories.
We holographically calculate two-point functions in the pseudo-conformal universe, an early universe alternative to inflation. The pseudo-conformal universe can be modeled as a defect conformal field theory, where the reheating surface is a codimension-1 spacelike defect which breaks the conformal algebra to a de Sitter subalgebra. The dual spacetime geometries are domain walls with de-Sitter symmetry in an asymptotically anti-de Sitter spacetime. We compute 2-point functions of scalars and stress tensors by solving the linearized equations for scalar and tensor fluctuations about these backgr…
We consider Einstein-Horndeski gravity with a negative bare constant as a holographic model to investigate whether a scale invariant quantum field theory can exist without the full conformal invariance. Einstein-Horndeski gravity can admit two different AdS vacua. One is conformal, and the holographic two-point functions of the boundary energy-momentum tensor are the same as the ones obtained in Einstein gravity. The other AdS vacuum, which arises at some critical point of the coupling constants, preserves the scale invariance but not the special conformal invariance due to the logarithmic rad…
Standard thermodynamic treatments of quantum field theory in the presence of black-hole backgrounds reproduce the black hole entropy by usually specializing to the leading order of the heat-kernel or the high-temperature expansion. By contrast, this work develops a hybrid framework centered on geometric spectral asymptotics whereby these assumptions are shown to be unwarranted insofar as black hole thermodynamics is concerned. The approach--consisting of the concurrent use of near-horizon and heat-kernel asymptotic expansions--leads to a proof of the holographic scaling of the entropy as a uni…
Logarithmic conformal field theories with vanishing central charge describe systems with quenched disorder, percolation or dilute self-avoiding polymers. In these theories the energy momentum tensor acquires a logarithmic partner. In this talk we address the construction of possible gravity duals for these logarithmic conformal field theories and present two viable candidates for such duals, namely theories of massive gravity in three dimensions at a chiral point.
We evaluate the holographic entanglement entropy, HEE, holographic mutual information, HMI, and holographic entanglement of purification, EoP, in a non-conformal model at zero and finite temperature. In order to find the analytical results we consider some specific regimes of the parameter space. We find that the non-conformal effects decrease the redefined HEE and increase the redefined HMI and EoP in the all studied regimes. On the other side, the temperature effects increase (decrease) the redefined HEE (HMI) in the all studied regimes while it has no definite effect on the redefined EoP. F…
In four lectures, delivered at the TASI 2010 summer school, I cover selected topics in the application of the gauge-string duality to nuclear and condensed matter physics. On the nuclear side, I focus on multiplicity estimates from trapped surfaces in AdS_5, and on the consequences of conformal symmetry for relativistic hydrodynamics. On the condensed matter side, I explain the fermion response to the zero-temperature limit of p-wave holographic superconductors.
We describe a holographic approach to explicitly compute the universal logarithmic contributions to entanglement and Renyi entropies for free conformal scalar and spinor fields on even-dimensional spheres. This holographic derivation proceeds in two steps: first, following Casini and Huerta, a conformal map to thermal entropy in a hyperbolic geometry; then, identification of the hyperbolic geometry with the conformal boundary of a bulk hyperbolic space and use of an AdS/CFT holographic formula to compute the resulting functional determinant. We explicitly verify the connection with the type-A …
We study corner contributions to holographic entanglement entropy in non-conformal backgrounds: a kink for D2-branes as well as a cone and two different types of crease for D4-branes. Unlike 2+1-dimensional CFTs, the corner contribution to the holographic entanglement entropy of D2-branes exhibits a power law behaviour rather than a logarithmic term. However, the logarithmic term emerges in the holographic entanglement entropy of D4-branes. We identify the logarithmic term for a cone in D4-brane background as the universal contribution under appropriate limits and compare it with other physica…
In this paper, we apply the K-theory scheme of classifying the topological insulators/superconductors to classify the topological classes of the massive multi-flavor fermions in anti-de Sitter (AdS) space. In the context of AdS/CFT correspondence, the multi-flavor fermionic mass matrix is dual to the pattern of operator mixing in the boundary conformal field theory (CFT). Thus, our results classify the possible patterns of operator mixings among fermionic operators in the holographic CFT.
We consider a holographic description of the inflationary multiverse, according to which the wave function of the universe is interpreted as the generating functional for a lower dimensional Euclidean theory. We analyze a simple model where transitions between inflationary vacua occur through bubble nucleation, and the inflating part of spacetime consists of de Sitter regions separated by thin bubble walls. In this model, we present some evidence that the dual theory is conformally invariant in the UV.
In this study, the gauge/string duality is used to investigate the dynamics of a moving heavy quark in a strongly coupled, non-conformal plasma. The drag force in this non-conformal model is smaller than that of $\mathcal{N} = 4$ supersymmetric Yang-Mills (SYM) plasma and decreases as the level of non-conformality is increased. At intermediate temperatures, the world-sheet temperature, which is derived numerically by calculating the world-sheet horizon, deviates from its conformal value, but at high temperatures, it tends to its conformal value.
For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulae for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmor…
In this paper, we investigate a simple holographic model which describes the conformal symmetry breaking at zero temperature. The model is implemented in the context of effective holographic models for QCD described by the Einstein-dilaton equations. The realization of spontaneous conformal symmetry breaking shows a massless state in the spectrum. The existence of this state is confirmed when we compute the two-point correlation function of the scalar operator in the dual field theory, where it has a pole. On the other hand, when there is an explicit conformal symmetry breaking the massless st…
The number of asymptotically de Sitter (non-singular) solutions of 5d dilatonic gravity with positive cosmological constant is found. These solutions are similar to the previously known asymptotically AdS spaces where dilaton may generate the singularity. Using these solutions the consistent $c$-function is proposed in the same way as in AdS/CFT. The consistency of RG flow gives further support for dS/CFT correspondence. From holographic RG flow equations we calculate the holographic 4d conformal anomaly with dilatonic contributions. This conformal anomaly turns out to be the same as in AdS/CF…
Recently, it is found that when an external magnetic field parallel to the boundary is applied, Weyl anomaly gives rises to a new anomalous current transport in the vicinity of the boundary. At the leading order of closeness from the boundary, the current is determined universally by the central charge of the theory. In this paper, we give a holographic proof for the existence and universality for this transport phenomena. We show that the current is independent of boundary conditions in four dimensions while it depends on boundary conditions in other dimensions. We also study the backreaction…
Significant work has gone into determining the minimal set of entropy inequalities that determine the holographic entropy cone. Holographic systems with three or more parties have been shown to obey additional inequalities that generic quantum systems do not. We consider a two dimensional conformal field theory that is a single boundary of a holographic system and find four additional non-linear inequalities which are derived from strong subadditivity and the formula for the entanglement entropy of a region on the conformal field theory. We also present an equality obtained by application of a…
I present two calculations of the holographic Weyl anomalies induced by Chern-Simons gravity theories alternative to the ones presented in the literature. The calculations presented here rest on the extension from Chern-Simons to Transgression forms as lagrangians, motivated by gauge invariance, which automatically yields the boundary terms suitable to regularize the theory. The procedure followed here sheds light in the structure of Chern-Simons gravities and their regularization.
We study second-order hydrodynamic transport in strongly coupled non-conformal field theories with holographic gravity duals in asymptotically anti-de Sitter space. We first derive new Kubo formulae for five second-order transport coefficients in non-conformal fluids in $(3+1)$ dimensions. We then apply them to holographic RG flows induced by scalar operators of dimension $Δ=3$. For general background solutions of the dual bulk geometry, we find explicit expressions for the five transport coefficients at infinite coupling and show that a specific combination, $\tilde{H}=2ητ_π-2(κ-κ^*)-λ_2$, al…
We report our recent results concerning d5 gauged supergravity (dilatonic gravity) considered on AdS background. The finite action on such background as well as d4 holographic conformal anomaly (via AdS/CFT correspondence) are found. In such formalism the bulk potential is kept to be arbitrary, dilaton dependent function. Holographic RG in such theory is briefly discussed. d5 AdS brane-world Universe induced by quantum effects of brane CFT is constructed. Such brane is spherical, hyperbolic or flat one. Hence, the possibility of quantum creation of inflationary brane-world Universe is shown.
The behaviors of the holographic superconductors/insulator transition are studied by introducing a charged scalar field coupled with a logarithmic electromagnetic field in both the Einstein-Gauss-Bonnet AdS black hole and soliton. For the Einstein-Gauss-Bonnet AdS black hole, we find that: i) the larger coupling parameter of logarithmic electrodynamic field $b$ makes it easier for the scalar hair to be condensated; ii) the ratio of the gap frequency in conductivity $ω_g$ to the critical temperature $T_c$ depends on both $b$ and the Gauss-Bonnet constant $α$; and iii) the critical exponents are…
We generalize the one-to one correspondence between quasi normal modes in 3- dimensional anti deSitter black holes and the poles of the retarded correlators in the boundary conformal field theory to include logarithmic operators in the latter. This analysis is carried out explicitly for the logarithmic mode in chiral topologically massive gravity.
We consider logarithmic extensions of the correlation and response functions of scalar operators for the systems with aging as well as Schrödinger symmetry. Aging is known to be the simplest nonequilibrium phenomena, and its physical significances can be understood by the two-time correlation and response functions. Their logarithmic part is completely fixed by the bulk geometry in terms of the conformal weight of the dual operator and the dual particle number. Motivated by recent experimental realizations of Kardar-Parisi-Zhang universality class in growth phenomena and its subsequent theoret…
In the AdS$_3$/CFT$_2$ correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Bañados geometry. We elaborate the constraints for a CFT state to be geometric, i.e., having a dual Bañados metric, by comparing the order of central charge of the entanglement/Rényi entropy obtained respectively from the holographic method and the replica trick in CFT. We find that the geometric CFT states fulfill Bohr's correspondence principle by reducing the quantum KdV hierarchy to its classical counterpart. We call the CFT states that satisfy the geometric constrai…
Three dimensional Gödel black hole is a solution to Einstein-Maxwell-Chern-Simons theory with a negative cosmological constant. We have studied the hidden conformal symmetry for massive scalar field without any additional condition in the background of three dimensional non-extremal and extremal Gödel black holes. This conformal symmetry is uncovered by the observation that the radial wave equations in both cases can all be rewritten in the form of $SL(2, R)$ Casimir operators through introducing two sets of conformal coordinates to write the $SL(2, R)$ generators. At last, we give the hologra…
We study holographic conformal order compactified on $S^3$. The corresponding boundary CFT$ _4$ has a thermal phase with a nonzero expectation value of a certain operator. The gravitational dual to the ordered phase is represented by a black hole in asymptotically $AdS_5$ that violates the no-hair theorem. While the compactification does not destroy the ordered phase, it does not cure its perturbative instability: we identify the scalar channel QNM of the hairy black hole with Im$[w]>0$. On the contrary, we argue that the disordered thermal phase of the boundary CFT is perturbatively stable in…
